University California Riverside (Department of Mathematics). IV Synthetic Projective Geometry

8/20/2019 University California Riverside (Department of Mathematics). IV Synthetic Projective Geometry

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CHAPTER IV

SYNTHETIC PROJECTIVE GEOMETRY

The purpose of this chapter is to begin the study of projective spaces, mainly from the synthetic point

of view but with considerable attention to coordinate projective geometry.

1. Axioms for projective geometry

The basic incidence properties of coordinate projective spaces are expressible as follows:Definition. A geometrical incidence space (S, Π, d) is projective if the following hold:

(P-1) : Every line contains at least three points.

(P-2) : If P and Q are geometrical subspaces of S then

d(P Q) = d(P ) + d(Q) − d(P ∩ Q) .

In particular, (P-2) is a strong version of the regularity condition (G-4) introduced in SectionII.5. The above properties were established for FPn (n ≥ 2) in Theorems III.10 and III.9respectively. It is useful to assume condition (P-1) for several reasons; for example, lines inEuclidean geometry have infinitely many points, and (P-1) implies a high degree of regularityon the incidence structure that is not present in general (compare Exercise 2 below and TheoremIV.11). — In this connection, note that Example 2 in Section II.5 satisfies (P-2) and every linein this example contains exactly two points.

Elementary properties of projective spaces

The following is a simple consequence of the definitions.

Theorem IV.1. If S is a geometrical subspace of a geometrical incidence space S , then S is a geometrical incidence space with respect to the subspace incidence structure of Exercise II.5.3.

If P is a projective incidence space and d(P ) = n ≥ 1, then P is called a projective n-space ; if

n = 2 or 1, then one also says that P is a projective plane or projective line , respectively.

Theorem IV.2. If P is a projective plane and L and M are distinct lines in P , then L ∩ M consists of a single point.

Theorem IV.3. If S is a projective 3-space and P and Q are distinct planes in S , then P ∩ Qis a line.

63


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64 IV. SYNTHETIC PROJECTIVE GEOMETRY

These follow from (P-2) exactly as Examples 1 and 2 in Section III.4 follow from Theorem III.9.

We conclude this section with another simple but important result:

Theorem IV.4. In the definition of a projective space, property (P-1) is equivalent to the following (provided the space is not a line):

(P-1) : Every plane contains a subset of four points, no three of which are collinear.

Proof. Suppose that (P-1) holds. Let P be a plane, and let X , Y and Z be noncollinearpoints in P . Then the lines L = X Y , M = X Z , and N = Y Z are distinct and contained in P .Let W be a third point of L, and let V be a third point of M .

Figure IV.1

Since L and M are distinct and meet at X , it follows that the points V , W , Y , Z must bedistinct (if any two are equal then we would have L = M ; note that there are six cases to check,with one for each pair of letters taken from W, X,Y,Z ). Similarly, if any three of these fourpoints were collinear then we would have L = M , and therefore no three of the points can becollinear (there are four separate cases that must be checked; these are left to the reader).

Conversely, suppose that (P-1) holds. Let L be a line, and let P be a plane containing L. Byour assumptions, there are four points A, B, C, D ∈ P such that no three are collinear.

Figure IV.2

Let M 1 = AB, M 2 = BC , M 3 = CD, and M 4 = AD. Then the lines M 1 are distinct andcoplanar, and no three of them are concurrent (for example, M 1 ∩ M 2 = M 3 ∩ M 4, and similarlyfor the others). It is immediate that M 1 contains at least three distinct points; namely, the


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1. AXIOMS FOR PROJECTIVE GEOMETRY 65

points A and B plus the point where M 1 meets M 3 (these three points are distinct because nothree of the lines M i are concurrent). Similarly, each of the lines M 2, M 3 and M 4 must containat least three points.

If L is one of the four lines described above, then we are done. Suppose now that L is not equalto any of these lines, and let P i be the point where L meets M i. If at least three of the pointsP 1, P 2, P 3, P 4 are distinct, then we have our three distinct points on L. Since no three of thelines M i are concurrent, it follows that no three of the points P i can be equal, and therefore if there are not three distinct points among the P i then there must be two distinct points, witheach P i equal to a unique P j for j = i. Renaming the M i if necessary by a suitable reordering of {1, 2, 3, 4}, we may assume that the equal pairs are given by P 1 = P 3 and P 2 = P 4. The drawingbelow illustrates how such a situation can actually arise.

Figure IV.3

We know that P 1 = P 3 and P 2 = P 4 are two distinct points of L, and Figure IV.3 suggests thatthe point Q where AC meets L is a third point of L. To prove this, we claim it will suffice toverify the following statements motivated by Figure IV.3:

(i) The point A does not lie on L.(ii) The line AC is distinct from M 1 and M 2.

Given these properties, it follows immediately that the three lines AC , M 1 and M 2 — which allpass through the point A which does not lie on L — must meet L in three distinct points (seeExercise 4 below).

Assertion (i) follows because A ∈ L implies

A ∈ M 2 ∩ L = M 4 ∩ M 2 ∩ L

and since A ∈ M 1 ∩ M 2 this means that M 1, M 2, and M 4 are concurrent at A. However, weknow this is false, so we must have A ∈ L. To prove assertion (ii), note that if AC = M 1 = AB,

then A , B , C are collinear, and the same conclusion will hold if AC = M 2 = BC . Since thepoints A , B , C are noncollinear by construction, it follows that (ii) must also hold, and asnoted above this completes the proof that L has at least three points.


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EXERCISES

1. Let (S, Π, d) be an n-dimensional projective incidence space (n ≥ 2), let P be a plane in S ,and let X ∈ P . Prove that there are at least three distinct lines in P which contain X .

2. Let n ≥ 3 be an integer, let P be the set {0, 1, · · · , n}, and take the family of subsetsL whose elements are {1, · · · , n} and all subsets of the form {0, k}, where k > 0. Show that(P, L) is a regular incidence plane which satisfies (P-2) but not (P-1). [Hint: In this case(P-2) is equivalent to the conclusion of Theorem IV.2.]

3. This is a generalization of the previous exercise. Let S be a geometrical incidence space of dimension n ≥ 2, and let ∞S be an object not belonging to S (the axioms for set theory giveus explicit choices, but the method of construction is unimportant). Define the cone on S tobe S • = S ∪ {∞S }, and define a subset Q of S

• to be a k-planes of S • if and only if eitherQ is a k-plane of S or Q = Q0 ∪ {∞S }, where Q0 is a (k − 1)-plane in S (as usual, a 0-planeis a set consisting of exactly one element). Prove that S • with these definitions of k-planes is ageometrical incidence (n + 1)-space, and that S • satisfies (P-2) if and only if S does. Explain

why S •

does not satisfy (P-1) and hence is not projective.4. Let (P, L) be an incidence plane, let L be a line in P , let X be a point in P which doesnot lie on L, and assume that M 1, · · · M k are lines which pass through X and meet L in pointsY 1, · · · Y k respectively. Prove that the points Y 1, · · · Y k are distinct if and only if the linesM 1, · · · M k are distinct.

5. Let (S, Π, d) be a regular incidence space of dimension ≥ 3, and assume that every plane inS is projective (so it follows that (P-1) holds). Prove that S is projective. [Hint: Since S isregular, condition (P-2) can only fail to be true for geometrical subspaces Q and R such thatQ ∩ R = ∅. If d(R) = 0, so that R consists of a single point, then condition (P-2) holds byTheorem II.30. Assume by induction that (P-2) holds whenever d(R0) ≤ k − 1, and supposethat d(R) = k. Let R0 ⊂ R be (k − 1)-dimensional, and choose y ∈ R such that y ∈ R 0. Showthat Q R0 ⊂ Q R, and using this prove that d(Q R) is equal to d(Q) + k or d(Q) + k + 1.The latter is the conclusion we want, so assume it is false. Given x ∈ Q, let x R denote the join of {x} and R, and define y Q similarly. Show that x R ∩ Q is a line that we shall callL, and also show that R ∩ y Q is a line that we shall call M . Since L ⊂ Q and M ⊂ R, itfollows that L ∩ M = ∅. Finally, show that x R ∩ y Q is a plane, and this plane containsboth L and M . Since we are assuming all planes in S are projective, it follows that L ∩ M isnonempty, contradicting our previous conclusion about this intersection. Why does this implythat d(Q R) = d(Q) + k + 1?]


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2. DESARGUES’ THEOREM 67

2. Desargues’ Theorem

In this section we shall prove a synthetic version of a fundamental result of plane geometrydue to G. Desargues (1591–1661). The formulation and proof of Desargues’ Theorem show that

projective geometry provides an effective framework for proving nontrivial geometrical theorems.

Theorem IV.5. (Desargues’ Theorem) Let P be a projective incidence space of dimension at least three, and let {A,B,C } and {A, B, C } be triples of noncollinear points such that the lines AA, B B and CC are concurrent at some point X which does not belong to either of {A,B,C }and {A, B, C }. Then the points

D ∈ BC ∩ BC

E ∈ AC ∩ AC

F ∈ AB ∩ AB

are collinear.

Figure IV.4

Proof. The proof splits into two cases, depending upon whether or not the sets {A,B,C } and{A, B, C } are coplanar. One feature of the proof that may seem counter-intuitive is that thenoncoplanar case is the easier one. In fact, we shall derive the coplanar case using the validityof the result in the noncoplanar case.


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CASE 1. Suppose that the planes determined by {A,B,C } and {A, B, C } are distinct. Thenthe point X which lies on AA, BB and CC cannot lie in either plane. On the other hand,it follows that the points {A, B, C } lie in the 3-space S determined by {X,A,B,C }, andtherefore we also know that the planes ABC and ABC are contained in S . These two planesare distinct (otherwise the two triples of noncollinear points would be the same), and hence their

intersection is a line. By definition, all three of the points D,E , F all lie in the intersection of the two planes, and therefore they all lie on the two planes’ line of intersection.

CASE 2. Suppose that the planes determined by {A,B,C } and {A, B, C } are identical. Theidea is to realize the given configuration as the photographic projection of a similar noncopla-nar configuration on the common plane. Since photographic projections onto planes preservecollinearity, this such a realization will imply that the original three points D ,E ,F are allcollinear.

Under the hypothesis of Case 2, all the points under consideration lie on a single plane we shallcall P . Let Y be a point not on P , and let Z ∈ AY be another. Consider the line A Z ; since A

and Z both lie on the plane AX Y , the whole line A Z lies in AX Y . Thus A Z and XY meet ina point we shall call Q.

Figure IV.5

Consider the following three noncoplanar triangle pairs:

(i) C QB and CY B.

(ii) C QA and CY A.

(iii) BQA and BY A.

Since AA, BB and CC all meet at X , the nonplanar case of the theorem applies in all threcases. Let G ∈ B Y ∩ B Q and H ∈ C Y ∩ C Q, and note that Z ∈ AR ∩ AQ. Then the truthof the theorem in the noncoplanar case implies that each of the triples

{D ,H ,G}, {F ,Y ,G}, {E ,H ,Z }

is collinear.


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2. DESARGUES’ THEOREM 69

Let P be the plane DF Z . Then G ∈ P by the collinearity of the second triple, and hence H ∈ P

by the collinearity of the first triple. Since Z, H ∈ P , we have E ∈ P by the collinearity of thethird triple. All of this implies that

E ∈ P ∩ P = DF

which shows that the set {D ,E ,F } is collinear.

Definition. A projective plane P is said to be Desarguian if Theorem 5 is always valid in P .

By Theorem 5, every projective plane that is isomorphic to a plane in a projective space of higherdimension is Desarguian. In particular, if F is a skew-field, then FP2 is Desarguian because FP2

is isomorphic to the plane in FP3 consisting of all points having homogeneous coordinates in F4

of the form (x1, x2, x3, 0). In Section 4 we shall note that, conversely, every Desarguian planeis isomorphic to a plane in a projective 3-space.

An example of a non-Desarguian projective plane (the Moulton plane )1 can be given by takingthe real projective plane RP2 as the underlying set of points, and modifying the definition of lines as follows: The new lines will include the line at infinity, all lines which have slope ≤ 0

or are parallel to the y -axis, and the broken lines defined by the equationsy = m(x − a), x ≤ a (i.e., y ≤ 0)

y = 12

m(x − a), x ≥ a (i.e., y ≥ 0)

where m > 0. As the points at infinite of the latter lines we take those belonging to y = m(x−a).A straightforward argument shows that the axioms for a projective plane are satisfied (seeExercise 4 below). However, as Figure IV.6 suggests, Desargues’ Theorem is false in this plane.

Figure IV.6

1Forest Ray Moulton (1872–1952) was an American scientist who worked mostly in astronomy but is also

recognized for his contributions to mathematics.


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Informally speaking, Desargues’ Theorem fails to hold for some projective planes because there isnot enough room in two-dimensional spaces to apply standard techniques. 2 Perhaps surprisingly,there are several other significant examples of geometrical problems in which higher dimensionalcases (say n ≥ N , where N depends upon the problem) are simpler to handle than lowerdimensional ones (see the paper of Gorn for another example).

Finally, we note that Theorems II.27–29 are basically special cases of Desargues’ Theorem.

EXERCISES

1. Explain why the results mentioned above are essentially special cases of Desargues’ Theorem.

2. Is it possible to “plant ten trees in ten rows of three?” Explain your answer using Desargues’Theorem.3

3. Explain why each of the following pairs of triangles in Euclidean 3-space satisfies thehypotheses of Desargues’ Theorem.

(i) Two coplanar triangles such that the lines joining the corresponding vertices are par-allel.

(ii) A triangle and the triangle formed by joining the midpoints of its sides.

(iii) Two congruent triangles in distinct planes whose corresponding sides are parallel.

4. Prove that the Moulton plane (defined in the notes) is a projective plane.

2For an example of a finite Non-Desarguian plane. see pages 158–159 of Hartshorne’s book.3For more information on the relevance of projective geometry to counting and arrangement problems, see

Section IV.3.


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3. DUALITY 71

3. Duality

By the principle of duality ... geometry is at one stroke [nearly] doubled in extent with no

expenditure of extra labor, — Eric Temple Bell (1883–1960), Men of Mathematics 4

Consider the following fundamental properties of projective planes:

(1) Given two distinct points, there is a unique line containing both of them.

(1∗) Given two distinct lines, there is a unique point contained in both of them.

(2) Every line contains at least three distinct points.

(2∗) Every point is contained in at least three distinct lines.

The important point to notice is that Statement (n∗) is obtained from Statement (n) by inter-changing the following words and phrases:

(i) point ←→ line

(ii) is contained in ←→ contains

Furthermore, Statement (n) is obtained from Statement (n∗) by exactly the same process. Sincethe four properties (1) − (1∗) and (2) − (2∗) completely characterize projective planes (seeExercise 1 below), one would expect that points and lines in projective planes behave somewhatsymmetrically with respect to each other.

This can be made mathematically precise in the following manner: Given a projective plane

(P, L), we define a dual plane (P ∗, L∗) such that P ∗ is the set L of lines in P and L∗ is in 1–1correspondence with P . Specifically, for each x ∈ P we define the pencil of lines with vertex xto be the set

p(x) = { L ∈ L = P ∗ | x ∈ L } .

In other words, L ∈ p(x) if and only if X ∈ L. Let P ∗∗ denote the set of all pencils associatedto the projective plane whose points are given by P and whose lines are given by P ∗.

Theorem IV.6. If (P, P ∗) satisfies properties (1) −(1∗) and (2) −(2∗), then (P ∗, P ∗∗) also does.

Proof. There are four things to check:

(a) Given two lines, there is a unique pencil containing both of them.

(a∗) Given two pencils, there is a unique line contained in both of them.

(b) Every pencil contains at least three lines.

(b∗) Every line is contained in at least three pencils.

4See the bibliography for more information and comments on this book.


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However, it is clear that (a) and (a∗) are rephrasings of (1∗) and (1) respectively, and likewise(b) and (b∗) are rephrasings of (2∗) and (2) respectively. Thus (1) − (1∗) and (2) − (2∗) for(P ∗, P ∗∗) are logically equivalent to (1) − (1∗) and (2) − (2∗) for (P, P ∗).

As indicated above, we call (P ∗, P ∗∗) the dual projective plane to (P, P ∗).

By Theorem 6 we can similarly define P ∗∗∗ to be the set of all pencils in P ∗∗, and it followsthat (P ∗∗, P ∗∗∗) is also a projective plane. However, repetition of the pencil construction doesnot give us anything new because of the following result:

Theorem IV.7. Let (P, P ∗) be a projective plane, and let p0 : P → P ∗∗ be the map sending

a point x to the pencil p(x) of lines through x. Then p0 defines an isomorphism of incidence planes from (P, P ∗) to the double dual projective plane (P ∗∗, P ∗∗∗).

Proof. By construction the map p0 is onto. It is also 1–1 because p(x) = p(y) implies everyline passing through x also passes through y. This is impossible unless x = y.

This it remains to show that L is a line in P if and only if p(L) is a line in P ∗∗

. But lines inP ∗∗ have the form

p(L) = { w ∈ P ∗∗ | L ∈ p(x) } .

where L is a line in P . Since L ∈ p(x) if and only if x ∈ L, it follows that p(x) ∈ p(L) if and onlyif x ∈ L. Hence p(L) is the image of L under the map p0, and thus the latter is an isomorphismof (projective) geometrical incidence planes.

The preceding theorems yield the following important phenomenon 5 which was described in thefirst paragraph of this section; it was discovered independently by J.-V. Poncelet (1788–1867)and J. Gergonne (1771–1859).

Metatheorem IV.8. (Principle of Duality) A theorem about projective planes remains true if one interchanges the words point and line and also the phrases contains and is contained in.

The justification for the Duality Principle is simple. The statement obtained by making theindicated changes is equivalent to a statement about duals of projective planes which correspondsto the original statement for projective planes . Since duals of projective planes are also projectiveplanes, the modified statement must also hold.

Definition. Let A be a statement about projective planes. The dual statement is the oneobtained by making the changes indicated in Metatheorem 8, and it is denoted by D(A) or A ∗.

EXAMPLE 1. The phrase three points are collinear (contained in a common single line) dualizesto three lines are concurrent (containing a common single point).

EXAMPLE 2. The property (P-1), which assumes the existence of four points, not threeof which are collinear, dualizes to the statement, There exist four lines, no three of which are concurrent. This statement was shown to follow from (P-1) in the course of proving Theorem4.

5This is called a Metatheorem because it is really a statement about mathematics rather than a theoremwithin mathematics itself. In other words it is a theorem about theorems .


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Figure IV.7

By construction, the lines B C , BC and DF all meet at E . Therefore Desargues’ Theoremapplies to the triples { C , C, F } and { B, B, D}, and hence we may conclude that the threepoints

X ∈ BB ∩ CC

G ∈ BD ∩ CF H ∈ BD ∩ C F

are collinear. Using this, we obtain the following additional conclusions:

(1) B, D ∈ γ , and therefore γ = BD.

(2) B, D ∈ γ , and therefore γ = B D.

(3) C, F ∈ β , and therefore β = C F .

(4) C , F ∈ β , and therefore β = C F .

Therefore G ∈ γ ∩ β and H ∈ γ ∩ β . Since the common points of these lines are A and A bydefinition, we see that X , A and A are collinear. In other words, we have X ∈ AA∩BB ∩CC ,and hence the three lines are concurrent, which is what we wanted to prove.


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3. DUALITY 75

Duality and finite projective planes

We shall illustrate the usefulness of duality by proving a few simple but far-reaching resultson projective planes which contain only finitely many points. We have already noted that foreach prime p there is a corresponding projective plane Z

pP2. As noted below, these are more

than abstract curiosities, and they play an important role in combinatorial theory (the study of counting principles) and its applications to experimental design and error-correcting codes.

By Theorem 9, it follows that a projective plane is finite if and only if its dual plane is finite.In fact, one can draw much stronger conclusions.

Theorem IV.11. Let P be a finite projective plane. Then all lines in P contain exactly the same number of points.

Proof. Let L and M be the lines. Since there exist four point, no three of which are collinear,there must exist a point p which belongs to neither L nor M .

Figure IV.8

Define a map f : L → M by sending x ∈ L to the point f (x) where px meets M . It is astraightforward exercise to verify that f is 1–1 and onto (see Exercise 6 below).

Dualizing the preceding, we obtain the following conclusion.

Theorem IV.12

. Let P be a finite projective plane. Then all points in P are contained in exactly the same number of lines.

Observe that the next result is self-dual, with the dual statement logically equivalent to theoriginal one.

Theorem IV.13. Let P be a finite projective plane. Then the number of points on each line is equal to the number of lines containing each point.


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Proof. Let L be a line in P and let x ∈ L. Define a map from lines through x to points on Lby sending a line M with x ∈ M to its unique intersection point with L. It is again a routineexercise to show this map is 1–1 and onto (again see Exercise 6 below).

Definition. The order of a finite projective plane is the positive integer n ≥ 2 such that every

line contains n + 1 points and every point lies on n + 1 lines. The reason for the subtracting onefrom the common number is as follows: If F is a finite field with q elements, then the order of FP2 will be equal to q .

The results above yield the following interesting and significant restriction on the number of points in a finite projective plane:

Theorem IV.14. Let P be a finite projective plane of order n. Then P contains exactly n 2+n+1points.

In particular, for most positive integers m it is not possible to construct a projective plane withexactly m points. More will be said about the possibilities for m below.

Proof. We know that n + 1 is the number or points on every line and the number of linesthrough every point. Let x ∈ P . If we count all pairs (y, L) such that L is a line through x andy ∈ L, then we see that there are exactly (n + 1)2 of them. In counting the pairs, some pointssuch as x may appear more than once. However, x is the only point which does so, for y = ximplies there is only one line containing both points. Furthermore, by the preceding result weknow that x appears exactly n + 1 times. Therefore the correct number of points in P is givenby subtracting n (not n + 1) from the number of ordered pairs, and it follows that P containsexactly

(n + 1)2 − n = n2 + n + 1

distinct points.

The next theorem follows immediately by duality.

Theorem IV.15. Let P be a finite projective plane of order n. Then P contains exactly n 2+n+1lines.

Further remarks on finite projective planes

We shall now consider two issues raised in the preceding discussion:

(1) The possible orders of finite projective planes.

(2) The mathematical and nonmathematical uses of finite projective planes.

ORDERS OF FINITE PROJECTIVE PLANES. The theory of finite fields is completely understoodand is presented in nearly every graduate level algebra textbook (for example, see Section V.5of the book by Hungerford in the bibliography). For our purposes it will suffice to note that foreach prime number p and each positive integer n, there is a field with exactly q = pn elements.It follows that every prime power is the order of some projective plane .


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3. DUALITY 77

The possible existence of projective planes with other orders is an open question. However,many possible orders are excluded by the following result, which is known as the Bruck–Ryser–Chowla Theorem: Suppose that the positive integer n has the form 4k + 1 or 4k + 2for some positive integer k. Then n is the sum of two (integral) squares .

References for this result include the books by Albert and Sandler, Hall (the book on grouptheory), and Ryser in the bibliography as well as the original paper by Bruck-Ryser (also in thebibliography) and the following online reference6:

http://www.math.unh.edu/∼dvf/532/7proj-plane.pdf

Here is a list of the integers between 2 and 100 which cannot be orders of finite projective planesby the Bruck-Ryser-Chowla Theorem:

6 14 21 22 30 33 38 42

45 46 54 57 62 66 69 70

74 77 78 82 86 93 94 97

The smallest positive integer ≥ 2 which is not a prime power and not excluded by the Bruck-Ryser-Chowla Theorem is 10. In the late nineteen eighties a substantial argument — whichused sophisticated methods together with involved massive amounts of computer calculations— showed that no projective planes of order 10 exist. Two papers on this work are listed in thebibliography; the article by one author (C. W. H. Lam) in the American Mathematical Monthly was written to explain the research on this problem to a reasonably broad general audience of mathematicians and students.

By the preceding discussion, the existence of projective planes of order n is understood forn ≤ 11, and the first open case is n = 12. Very little is known about this case.

PROJECTIVE GEOMETRY AND FINITE CONFIGURATIONS. Of course, one can view existenceproblems about finite projective planes as extremely challenging puzzles similar to Magic Squares(including the Sudoku puzzles that have recently become extremely popular), but one importantreason for studying them is their relevance to questions of independent interest.

One somewhat whimsical “application” ( plant ten trees in ten rows of three ) was mentioned inExercise IV.2.2, where the point was that such configurations exist by Desargues’ Theorem. Infact, the study of projective spaces — especially finite ones – turns out to have many usefulconsequences in the study of finite tactical configurations or block designs , which is part of combinatorial theory or combinatorics . Specifically, finite projective planes are of interest asexamples of Latin squares , or square matrices whose entries are in a finite set such that eachelement appears in every row and every column exactly once. As noted earlier, such objects playa significant role in the areas of statistics involving the design of experiments and in the theory of error-correcting codes. Although further comments are well outside the scope of these notes, thereferences in the bibliography by Buekenhout, Crapo and Rota, Hall (the book on combinatorialtheory), Kárteszi, Lindner and Rodgers, and (the previously cited book by) Ryser are all sourcesfor further information. There are also several online web sites dedicated to questions aboutfinite geometry.

6To update this document, the conjecture of E. Catalan (1814–1894) has recently been shown to be truebu P. Mihăilescu. The proof is at a very advanced level, but for the sake of completeness here is a reference:P. Mihăilescu, Primary cyclotomic units and a proof of Catalan’s Conjecture , [Crelle] Journal für die reine undangewandte Mathematik 572 (2004), 167–195.


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Duality in higher dimensions

The concept of duality extends to all projective n-spaces (where n ≥ 2), but the duality is morecomplicated for n ≥ 3 than it is in the 2-dimensional case. Specifically, if S is a projectiven-space, then points of the dual space S ∗ are given by the hyperplanes of S . for each k-plane

P in S , there is an associated (linear) bundle of hyperplanes with center P , which we shalldenote by b(P ), and the dimension of b(P ) is set equal to n − k − 1. We shall also denote theset of all such bundles by Π∗ the associated dimension function by d∗. The following result isthe appropriate generalization of Theorems 6 and 7 which allows extension of the principle of duality to higher dimensions.

Theorem IV.16. If (S, Π, d) is a projective n-space for some n ≥ 2, then so is the dual object (S ∗, Π∗, d∗). Furthermore, the map E sending x ∈ S to the bundle of hyperplanes b(x) with center x defines an isomorphism of geometrical incidence spaces.

We shall not give a direct proof of this result for two reasons.

1. Although the proof is totally elementary, it is a rather long and boring sequence of routine verifications.

2. The result follows from a coordinatization theorem in the next section (Theorem 18)and the results of Section VI.1, at least if n ≥ 3 (and we have already done the casen = 2).

An excellent direct proof of Theorem 16 is given in Sections 4.3 and 4.4 of Murtha and Willard,Linear Algebra and Geometry (see the bibliography for further information).

EXERCISES

1. Prove that properties (1) − (1∗) and (2) − (2∗) completely characterize projective planes. Inother words, if (P, L) is a pair consisting of a set P and a nonempty collection of proper subsetsL satisfying these, then there is a geometrical incidence space structure (P, L, d) such that theincidence space is a projective plane and L is the family of lines in Π.

2. Write out the plane duals of the following phrases, and sketch both the given data and theirplane duals.

(i) Two lines, and a point on neither line.

(ii) Three collinear points and a fourth point not on the line of the other three.

(iii) Two triples of collinear points not on the same line.(iv) Three nonconcurrent lines, and three points such that each point lies on exactly one of

the lines.


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3. DUALITY 79

3. Draw the plane duals of the illustrated finite sets of points which are marked heavily in thetwo drawings below.

Figure IV.9

4. Suppose that f : S → T is an incidence space isomorphism from one projective n-space(n ≥ 2) to another. Prove that f induces a 1–1 correspondence f ∗ : S ∗ → T ∗ taking a hyperplaneH ⊂ S to the image hyperplane f ∗(H ) = f [H ] ⊂ T . Prove that this correspondence has the

property that B is an r-dimensional bundle of hyperplanes (in S ∗) if and only if f ∗[B] is sucha subset of T ∗ (hence it is an isomorphism of geometrical incidence spaces, assuming Theorem16).

5. Prove that the construction in the preceding exercise sending f to f ∗ has the followingproperties:

(i) If g : T → U is another isomorphism of projective n-spaces, then (g of )∗ = g∗ o f ∗.

(ii) For all choices of S the map (idS )∗ is equal to the identity on S ∗.

(iii) For all f we have

f −1 ∗

=

f ∗−1

.

6. Complete the proof of Theorem 11.

7. Let (P, L) be a finite affine plane. Prove that there is a positive integer n such that

(i) every line in P contains exactly n points,

(ii) every point in P lies on exactly n lines,

(iii) the plane P contains exactly n2 points.


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4. Conditions for coordinatization

Since the results and techniques of linear algebra are applicable to coordinate projective n-spacesS 1(V ) (where dim V = n + 1), these are the most conveniently studied of all projective spaces.

Thus it is desirable to know when a projective n-space is isomorphic to one having the formS 1(V ), where dim V = n + 1. The following remarkable theorem shows that relatively weakhypotheses suffice for the existence of such an isomorphism.

Theorem IV.17. Let P be a projective n-space in which Desargues’ Theorem is valid (for exam-ple, n ≥ 3 or P is a Desarguian plane). Then there is a skew-field F such that P is isomorphic to FPn (where we view Fn+1 as a right vector space over F. If E is another skew-field such that P is isomorphic to EPn, then E and F are isomorphic as skwe-fields.

A well-illustrated proof of Theorem 17 from first principles in the case n = 2 appears on pages175–193 of the book by Fishback listed in the bibliography. Other versions of the proof appearin several other references from the bibliography. Very abstract approaches to the theorem whenn = 2 appear in Chapter III of the book by Bumcrot and also in the book by Artzy. The proof in Chapter 6 of Hartshorne’s book combines some of the best features of the other proofs. Thereis also a proof of Theorem 17 for arbitrary values of n ≥ 2 in Chapter VI from Volume I of Hodge and Pedoe. Yet another reference is Sections 4.6 and 4.7 of Murtha and Willard. Formore information on the uniqueness statement, see Theorem V.10.

Theorem 17 yields a classification for projective n-spaces (n ≥ 3) that is parallel to TheoremII.38 (see Remark 3 below for further discussion). Because of its importance, we state thisclassification separately.

Theorem IV.18. Let P be a projective n-space, where n ≥ 3. Then there is a skew-field

F, unique up to algebraic isomorphism, such that P and FPn

are isomorphic as geometrical incidence spaces.

Theorem 17 also implies the converse to a remark following the definition of a Desarguianprojective plane in Section IV.2.

Theorem IV.19. If a projective plane is Desarguian, then it is isomorphic to a plane in a projective 3-space.

Proof. By Theorem 17, the plane is isomorphic to FP2 for some skew-field F. But FP2 isisomorphic to the plane in FP3 defined by x4 = 0.

REMARK 1. Suppose that P is the Desarguian plane FP2. By Theorem 10, P ∗ is also Desarguianand hence is isomorphic to some plane EP2. As one might expect, the skew-fields F and E areclosely related. In fact, by Theorem V.1 the dual plane P ∗ is isomorphic to S 1(F3), where F3 isa left vector space over F, and left vector spaces over F correespond to right vector spaces overthe opposite skew-field FOP, whose elements are the same as F and whose multiplication isgiven by reversing the multiplication in F. More precisely, one defines a new product ⊗ in Fvia a ⊗ b = b · a, and if V is a left vector space over F define vector space operations via thevector addition on F and the right scalar product x ⊗ a = a · x (here the dot represents the


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4. CONDITIONS FOR COORDINATIZATION 81

original multiplication). It is a routine exercise to check that FOP is a skew-field and ⊗ makesleft F-vector spaces into right vector spaces over FOP. Thus, since F3 is a 3-dimensional rightFOP-vector space, it follows that E must be isomorphic to FOP.

REMARK 2. If P is the coordinate projective plane FP2 and multiplication in F is commutative,then the preceding remark implies that P and P ∗ are isomorphic because F and FOP are identicalin such cases. However, the reader should be warned that the isomorphisms from P to P ∗ aremuch less “natural” than the standard isomorphisms from P to P ∗∗ (this is illustrated byExercise V.1.5). More information on the noncommutative case appears in Appendix C.

EXAMPLE 3. Theorem II.18 follows from Theorem 18. For if S is an affine n-space, let S be its synthetic projective extension as defined in the Addendum to Section III.4. By TheoremIII.16, we know that S is an n-dimensional projective incidence space, so that S is isomorphicto FPn for some skew-field F and S ⊂ S is the complement of some hyperplane. Since there isan element of the geometric symmetry group of FPn taking this hyperplane to the one definedby xn+1 = 0 (see Exercise III.4.14), we can assume that S corresponds to the image of F

n inFP

n. Since k-planes in S are given by intersections of k-planes in S with S and similarly thek-planes in Fn are given by intersections of k-planes in FPn with the image of F, it follows thatthe induced 1–1 correspondence between S and Fn is a geometrical incidence space isomorphism.

A general coordinatization theory for projective planes that are not necessarily Desarguianexists; the corresponding algebraic systems are generalizations of skew-fields known as planar ternary rings . Among the more accessible references for this material are the previously citedbook by Albert and Sandler, Chapter 4 of the book by Artzy, Chapters III and VI of the bookby Bumcrot, and Chapter 17 of the book by Hall on group theory. One of the most importantexamples of a planar ternary ring (the Cayley numbers) is described on pages 195–196 of Artzy7 As one might expect, the Cayley numbers (also called octonions ) are associated to aprojective plane called the Cayley projective plane .

7One brief and accurate online summary is given in http://en.wikipedia.org/wiki/Octonion. The relia-bility issue for Wikipedia articles is discussed in a footnote for Appendix C, and it also applies here.


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University California Riverside (Department of Mathematics). IV Synthetic Projective Geometry